Domatic Number of a Graph and its Variants (Extended Abstract)

2015, 1992
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Elsevier Science Bv
Technical university of Liberec, Czech Republic
Technická Univerzita v Liberci
This chapter presents some numerical invariants of graphs that are related to the concept of domination—namely, the domatic number and its variants.. The word domatic was coined from the words dominating and chromatic in the same way as the word smog was composed from the words smoke and fog. This concept is a certain analogy of the chromatic number, but instead of independent sets, dominating sets are used in its definition. A subset D of the vertex set V(G) of an undirected graphs G is called dominating if for each x V(G) − D there exists a vertex yD adjacent to x. A domatic partition of G is a partition of V(G), all of whose classes are dominating sets in G. The maximum number of classes of a domatic partition of G is called the “domatic number” of G and denoted by d(G). R. Laskar and S. T. Hedetniemi have introduced the connected domatic number d, (G) of a graph G. It is the maximum number of classes of a partition of V(G) into dominating sets that induce connected subgraphs of G.